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Theorem naryrcl 48481
Description: Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)
Hypothesis
Ref Expression
naryfval.i 𝐼 = (0..^𝑁)
Assertion
Ref Expression
naryrcl (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))

Proof of Theorem naryrcl
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-naryf 48477 . 2 -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛m (𝑛m (0..^𝑥))))
21elmpocl 7674 1 (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  (class class class)co 7431  m cmap 8865  0cc0 11153  0cn0 12524  ..^cfzo 13691  -aryF cnaryf 48476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-naryf 48477
This theorem is referenced by:  naryfvalelfv  48482  0aryfvalelfv  48485  fv1arycl  48487  1arymaptfv  48490  fv2arycl  48498  2arymaptfv  48501
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